1
algebra

How many solutions does this system have? {6x3y=122xy=4\begin{cases} 6x - 3y = 12 \\ 2x - y = 4 \end{cases}

A

Exactly one solution

B

No solution

C

Infinitely many solutions

D

Cannot be determined

Correct Answer: C

Choice C is the correct answer. Let's check if one equation is a multiple of the other.

Step 1: Divide the first equation by 3: 6x3y3=123\frac{6x - 3y}{3} = \frac{12}{3}2xy=42x - y = 4

This is exactly the second equation!

Step 2: Since both equations represent the same line, every point on that line satisfies both equations.

Conclusion: Infinitely many solutions

💡 Strategic Tip: When equations are scalar multiples of each other (including the constant), they represent the same line.

Choice A is incorrect because one solution requires different slopes.

Choice B is incorrect because no solution requires parallel lines with different constants.

Choice D is incorrect because we can always determine this by simplifying.